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Theorem erbr3b 28807
Description: Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.)
Assertion
Ref Expression
erbr3b ((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))

Proof of Theorem erbr3b
StepHypRef Expression
1 simpll 786 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝑅 Er 𝑋)
2 simplr 788 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐵)
3 simpr 476 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐴𝑅𝐶)
41, 2, 3ertr3d 7647 . 2 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐴𝑅𝐶) → 𝐵𝑅𝐶)
5 simpll 786 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝑅 Er 𝑋)
6 simplr 788 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐵)
7 simpr 476 . . 3 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶)
85, 6, 7ertrd 7645 . 2 (((𝑅 Er 𝑋𝐴𝑅𝐵) ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
94, 8impbida 873 1 ((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   class class class wbr 4583   Er wer 7626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-er 7629
This theorem is referenced by: (None)
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